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Properties

Label 2-637-13.9-c1-0-18
Degree $2$
Conductor $637$
Sign $0.872 + 0.488i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.5 + 2.59i)3-s + (0.500 + 0.866i)4-s − 3·5-s + (−1.5 − 2.59i)6-s − 3·8-s + (−3 − 5.19i)9-s + (1.5 − 2.59i)10-s + (1.5 − 2.59i)11-s − 3·12-s + (1 + 3.46i)13-s + (4.5 − 7.79i)15-s + (0.500 − 0.866i)16-s + (−1 − 1.73i)17-s + 6·18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 + 1.49i)3-s + (0.250 + 0.433i)4-s − 1.34·5-s + (−0.612 − 1.06i)6-s − 1.06·8-s + (−1 − 1.73i)9-s + (0.474 − 0.821i)10-s + (0.452 − 0.783i)11-s − 0.866·12-s + (0.277 + 0.960i)13-s + (1.16 − 2.01i)15-s + (0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + 1.41·18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-1 - 3.46i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.5 + 11.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80761314769465912166222247823, −9.319309177067364670683997106943, −8.953065874571569495482158931145, −7.926414403291826510603158269533, −6.89395633091701794443273327264, −6.08453405277062383756315652768, −4.89627513052537269191172363045, −3.91578565359133822835240984392, −3.33453984966480936188307050410, 0, 1.18585911043870389244667343960, 2.39667627224644028421078766268, 3.94114065870228253874817188542, 5.46005584618270116139243791811, 6.26385727754423665528886900574, 7.20892396848169732998825106118, 7.79567633373124016427401213919, 8.782157788270012433070236009096, 10.11119986112905739488476531203

Graph of the $Z$-function along the critical line